Almost periodicity of some error terms in prime number theory

ACTA ARITHMETICA 106.3 (2003) Almost periodicity of some error terms in prime number theory by Jerzy Kaczorowski (Poznan) and Olivier Ramare (Lille) 1. Introduction and statement of results. The aim of this paper is to investigate distribution of values of a large class of functions of arithmetic significance assuming a suitably generalized Riemann Hypothesis. Probably the simplest example of a member of this class is defined by the following formula: ?0(v) = ? ?? ?? e?v/2 ( ? ?0(ev) + ev ? 12 log(1? e?2v)? log 2pi ) if v > 0, e?v/2 ( ?˜0(e?v) + ev + v + 1 2 log 1? ev 1 + ev + C ) if v < 0, (1) where C is the Euler constant and as usual (for x > 1) ?(x) = ∑ n≤x ?(n), ?˜(x) = ∑ n≤x ?(n)/n, ?0(x) = 12 (?(x+ 0) + ?(x? 0)), ?˜0(x) = 12(?˜(x+ 0) + ?˜(x? 0)). This function for positive v is only a mild modification of the normalized remainder term in the prime number formula, where we take the effect of the trivial zeros of the zeta function into account.

• riemann zeta


• generalized riemann


• indicated half-planes


• selberg class


• arithmetic pro


• function ?


• class contains most